- What is the radius of a confidence interval?
- What sample size will provide a given margin of error?
- What is the level of confidence of a given interval?

Another problem asks how large n must be to get a confidence
interval of a specified radius. (Recall that the larger n is, the smaller the
radius of the confidence interval is.

Example: If *sigma* = 15, how large must n be to get a 90% confidence
interval with radius less than or equal to .5?

90% confidence provides *alpha*/2 = .05; looking up .95 in the body of the
normal table gives z_.05=1.65. Therefore we want to solve
.5 > 1.65 × 15/(n^.5) which is equivalent to n^.5 > 1.65 ×
15/.5 = 49.5. Squaring both sides provides n must be greater than 2450.25,
or at least 2451 since n is an integer.

Sometimes, one may have collected the data, and wonder how
confident he can
be that the mean is within a given distance of x-bar.

Example: If x-bar = 39 based on a sample of size 60, and *sigma* = 20, how
confident are you that the interval (35, 43) contains µ?

Since 4 is the radius of the confidence interval, 4=z_(*alpha*/2) ×
*sigma*/(n^.5). Since *sigma*=20 and n=60, this becomes 4=z_(*alpha*/2)
× 20/(60^.5), which can be solved for z_(*alpha*/2) = 1.5492. From the
normal table we find *alpha*/2=.0606, hence the level of confidence,
(1-*alpha*), is .8788.

**Competencies:** If the standard deviation of your population is 13, how large must the sample size be to get a 95% confidence interval with a margin of error (radius) less than or equal to 2?

If the standard deviation of your population is 12 and based on a sample of size 60 you get the confidence interval (17, 22), what is the level of confidence of that interval (recall that x-bar is in the middle of the confidence interval).